Anomaly-induced Thermodynamics in Higher Dimensional AdS/CFT Gim - PowerPoint PPT Presentation

Anomaly-induced Thermodynamics in Higher Dimensional AdS/CFT Gim Seng Ng McGill University Northeast Gravity Workshop, 23rd April 2016

Based on : 1311.2940, 1407.6364 and 1505.02816 
 (with T. Azeyanagi, R. Loganayagam and M. J. Rodriguez)

Motivations BHs have entropy ! Entropy-matching … Often extremal or/and susy… How about non-extremal/ non-SUSY finite temperature entropy matching? BTZ entropy is reproduced by the universal Cardy’s formula Higher-dimensional generalizations of Cardy’s formula? Higher-dimensional AdS/CFT “Cardy entropy-matching”?

Outline Chiral Half of Cardy’s Formula in ��� 2 n Replacement Rule from ��� 2 n + 1 ���� 2 n

chiral half of Cardy Consider a 2d CFT on a circle of radius R Let us put it at finite temperature and rotation/boost At high temperature: T � 1 / R Time 2 πR 2 Ω 2 πR � c R + c L � c R − c L � � S Cardy ≈ ( 4 πT ) ( 4 πT ) + 1 − R 2 Ω 2 1 − R 2 Ω 2 24 24 R Weyl Anomaly Gravitational Anomaly (``Chiral Half’’/anomalous-part)

chiral HALF of Cardy 2 πR 2 Ω � c R − c L � S CFT 2 , anom ≈ ( 4 πT ) 1 − R 2 Ω 2 24 Generalizations to higher-d CFT (on sphere) Replacement rule [Surowka, Loganayagam, Jensen, Yarom,…] To understand the replacement rule, we need to review the following two things: T=0 anomalies Anomalous hydrodynamics

ANOMAL Y INFLOW [Callan and Harvey] Anomalies are captured by Chern-Simons terms Append an extra auxiliary direction. I CS The (2n+1) theory is anomaly-free, but with Chern-Simons terms Non-conservation of the (2n)-theory is captured by the ``inflow’’ of charges into the extra auxiliary direction: � μ J μ | QFT 2 n � j ⊥ j ⊥ j ⊥ ∼ �� P anom P anom = dI CS � F QFT 2 n P anom �� � ���������� �� F ��� R

���������� ������ anomalous hydro Hydro: effective long-wavelength description: �������� ���������� { u α , T , μ , . . . } � � A α , g αβ , . . . Hydro derivative expansion: J α = qu α + . . . +( J α ) anom + . . . � � T αβ = Eu α u β + p g αβ + u α u β + . . . +( T αβ ) anom + . . . Leading anomalous contribution: V α � ε αβ ... u β ( � u ) n − 1 parity-odd vorticity : ...

����� anomalous hydro J α = qu α + . . . +( J α ) anom + . . . � � T αβ = Eu α u β + p g αβ + u α u β + . . . +( T αβ ) anom + . . . ( J α ) anom = − ∂ F [ T , μ ] V α + . . . ∂ μ � � � ∂ F � � ∂ F � � � ( T αβ ) anom = F − μ − T u α V β + V α u β + . . . ∂ μ ∂ T T μ � � F � � � F � � � CFT V μ dx μ = − � ( 2 πR 2 Ω i ) S anom = − � T � T μ μ i is like an anomalous Gibbs free energy F Recall that standard relations: Q = − ∂ G � ∂ G � ∂ G � ∂ G � � � E = G − μ − T S = − ∂ μ , ∂ μ ∂ T ∂ T , μ μ T

����� anomalous hydro ( J α ) anom = − ∂ F [ T , μ ] V α + . . . ∂ μ � � � ∂ F � � ∂ F � � � ( T αβ ) anom = F − μ − T u α V β + V α u β + . . . ∂ μ ∂ T μ T � � F � � � F � � � CFT V μ dx μ = − � ( 2 πR 2 Ω i ) S anom = − � T � T μ μ i Question: What is ? F

chiral HALF of Cardy Example: 2d CFT Cardy’s formula � c R − c L � S CFT 2 , anom ≈ 2 πR 2 Ω ( 4 πT ) 24 S anom = − ∂ F ∂ T ( 2 πR 2 Ω ) F = − c R − c L 2 ( 2 π ) 242 ( 2 πT ) 2 = c g �� [ R 2 ] � � trR 2 → 2 ( 2 πT ) 2 anomaly polynomial for 2d grav. anomaly

����� Replacement rule OR chiral half of cardy’s formula [Surowka, Loganayagam, Jensen, Yarom,…] V α � ε αβ ... u β ( � u ) n − 1 ( J α ) anom = − ∂ F [ T , μ ] V α + . . . ∂ μ � � � ∂ F � � ∂ F � � � ( T αβ ) anom = F − μ − T u α V β + V α u β + . . . ∂ μ ∂ T T μ � � F � � � F � � � CFT V μ dx μ = − � ( 2 πR 2 Ω i ) S anom = − � T � T μ μ i F = P anom [ F , R ] | F → μ , �� [ R 2 k ] → 2 ( 2 πT ) 2 k

����� REPLACEMENT RULE 2d: P anom = c A F 2 + c g �� [ R 2 ] , F = c A μ 2 + c g 2 ( 2 πT ) 2 4d: P anom = c A F 3 + c M F �� [ R 2 ] , F = c A μ 3 + c M μ × 2 ( 2 πT ) 2 6d: P anom � c g �� [ R 4 ] , F = c g 2 ( 2 πT ) 4 ( J α ) anom = − ∂ F [ T , μ ] V α + . . . ∂ μ � � � ∂ F � � ∂ F � � � ( T αβ ) anom = F − μ − T u α V β + V α u β + . . . ∂ μ ∂ T T μ � � F � � � F � � � CFT V μ dx μ = − � ( 2 πR 2 Ω i ) S anom = − � T � T μ μ i

Outline Chiral Half of Cardy’s Formula in ��� 2 n Replacement Rule from 
 ��� 2 n + 1 ���� 2 n - Testing AdS/CFT in the presence of anomalies 
 - What bulk geometric structure captures the 
 replacement rule?

A holographer ’s recipe 1. Find your favorite AdS bulk theories Issue 1: Need gauge-gravitational Chern-Simons terms 2. Find the relevant AdS BH solutions Issue 2: Need AdS-Kerr-Newman+CS solutions 
 (with all rotations/charges turned on) 3. Calculate responses, charges, thermodynamics… Issue 3: Holographic renormalization/charges / entropy for Chern-Simons terms 4. Match/predict CFT results

step 1: favorite Ads gravity setup Toy model: D=2n+1 Einstein-Maxwell+negative c.c. 
 +CS terms Ics Equations of motion: R ab − 1 � � 2 ( R − 2Λ ) g ab = 8 πG N ( T M ) ab + ( T H ) ab � b F ab = g 2 YM ( J H ) a ( T M ) ab Maxwell contribution: CS/``Hall’’ contributions: � � P anom � J H = − � � F � � P anom � ( T H ) ab = � c Σ ( ab ) c Σ ( ab ) c = − 2 � P anom = dI CS � R ab

step 2: fluid/gravity BH solution (no anomalies) Gravity dual of charged rotating fluid 
 (for now, without anomalies) ds 2 = − 2 u μ dx μ dr + r 2 � dx μ dx ν + ��� ����� � − f ( r , m , q ) u μ u ν + P μν A = Φ ( r , q ) u μ dx μ + ��� ����� P μν = g μν + u μ u ν Non-trivial bulk radial dependence: q q 2 f ( r , m , q ) = 1 − m r 2 n + 1 Φ ( r , q ) = 2 κ q r 2 n − 2 r 2 ( 2 n − 1 ) Horizon values: Φ T ( r ) ≡ r 2 df Φ ( r H ) = μ , Φ T ( r H ) = 2 πT 2 dr

step 2: fluid/gravity BH solution (with anomalies) Gravity dual of anomalous charged rotating fluid ds 2 = − 2 u μ dx μ dr + r 2 � dx μ dx ν + ��� ����� � − f ( r , m , q ) u μ u ν + P μν + g V ( r , m , q )( u μ V ν + u ν V μ ) dx μ dx ν + . . . A = Φ ( r , q ) u μ dx μ + ��� ����� R ab − 1 2 ( R − 2Λ ) g ab = 8 πG N � ( T M ) ab + ( T H ) ab � + a V ( r , m , q ) V μ dx μ + . . . � b F ab = g 2 YM ( J H ) a g V , a V Leading contributions from the CS-terms : 
 Bulk replacement rule: � ∂ G � ∂ G � Φ T ( r ) ≡ r 2 � df T H ∼ ∂ 2 J H ∼ ∂ r 2 dr r ∂ Φ ∂ Φ T � � F → Φ , �� [ R 2 k ] → 2Φ 2 k G ≡ P anom T

step 3&4: currents and replacement rule Bulk replacement rule metric corrections boundary stress-tensor/currents G − Φ ∂ G ∂ G � � � � � � � � T αβ T ���������� ∂ Φ − Φ T u α V β + u β V α anom = anom = αβ ∂ Φ T r = r H � ∂ G � ( J α ) anom ∼ g μα ( F rμ ) anom | �������� = − V α ∂ Φ r = r H G ( r = r H ) = F [ F → μ , �� [ R 2 k ] → 2 ( 2 πT ) 2 k ] Bulk replacement rule-> boundary replacement rule !!

����� ������� step 3&4: entropy Tachikawa Formula (see also Solodukhin and Bonora-Cvitan-Prester-Pallua-Smolic) 
 (note: NOT Wald’s formula applied to Ics) ∞ ∂ P anom � ( 8 πk ) Γ N R 2 k − 2 � S CS = N ∂ �� [ R 2 k ] k = 1 P anom = dI cs , Γ N , R N = ������ ������ �������������������� Applying this to the fluid/gravity metric we found gives � � F � � � F � � � CFT V μ dx μ = − � ( 2 πR 2 Ω i ) S anom = − � T � T μ μ i Computations are opaque, long and tedious … 
 (what bulk structures imply the replacement for entropy?)

summary, current and future work Higher-dimensional chiral-half of Cardy’s formula 
 (Replacement rule) Construct (anomalous) fluid/gravity solutions in the bulk for the Einstein-Maxwell-CS theory Found ``bulk replacement rule’’ which implies the boundary replacement rule Current/Future: Time-dependence (non-stationary) and/or higher-order? More realistic AdS/CFT setup … add matter and etc 
 (should not alter the conclusions) Anomaly-induced entanglement entropy [Azeyanagi-Loganayagam-Ng, 1507.02298]